We show that a deformable mirror with four point actuators is suitable for the simultaneous correction of astigmatism and defocus in a given optical system by adequately choosing the relative position of the actuators. An analytical model is developed that describes adequately the mirror deformation as a function of actuator position, showing that it is possible to continuously tune the weight of each aberration. Experimental measurements with a single adjuster deformable mirror assembly confirm the validity of the model.
© 2007 Optical Society of America
The generation of ultraintense optical pulses nowadays relies on focusing high power, ultrashort laser pulses from solid-state chirped-pulse amplification systems to a near diffraction-limited spot size. Recently, a focused intensity of magnitude 1022 W/cm2 was obtained by a careful control and adaptive correction of the optical aberrations in a 45-TW Ti:sapphire laser . In fact, optical aberrations are a major problem in a large laser chain, degrading the focal spot quality and dramatically decreasing the Strehl ratio and the peak focused intensity. In multiterawatt to petawatt-level Nd:glass-based laser systems, the main aberrations present can be of a static nature, resulting from the propagation inside optical elements such as grating compressors, or dynamical, arising from thermal effects in the rod and disk amplifiers, in particular thermal lensing and thermal birefringence astigmatism [2, 3, 4]. These effects, apart from inducing serious wavefront distortions, limit the repetition rate to a few shots per hour at best.
Several approaches have been used to compensate both static and dynamical aberrations in these laser systems, such as bimorph deformable mirrors (DM) coupled to a feedback loop [2, 5, 6], single-actuator mechanically DM for low order correction of thermal lensing [7, 8] and cylindrical aberration  and a four actuator DM for astigmatism correction . While bimorph DM can provide correction for virtually any aberration, compared to the specific purpose of the mechanical type, their can be less cost-effective, particularly for large optical apertures, if the aberrations to correct fall within the most frequent ones.
To address this issue, we have developed a single actuator DM assembly with four contact points, and we demonstrate that it is suitable for the simultaneous correction of astigmatism and defocus, the most frequent aberrations in high power lasers. The balance between both aberrations can be controlled by the relative position of the four contacts.
2. Analytical model
Let us consider a flat, circular substrate of thickness h, less than half its radius a, simply supported at two ends of a diameter, and subjected to a pair of point loads P/2 at a distance b from the center, along the perpendicular diameter, as represented in Fig. 1. For small transverse deflections, we may use an analytical description for the deflected profile. According to classical plate theory , we have ∇∇w(r,θ) = P/D, where w(r,θ) is the deflection along z in polar coordinates, P is the total load applied, and D = Eh 3/(12(1-v 2)) is the flexural rigidity of the substrate, and E = 82×109 Pa and v = 0.206 are respectively Young’s modulus and Poisson’s ratio for BK7. The solution of this equation is of the form w(r,θ) = w 1(r,θ)+w 0(r,θ), where w 1(r,θ) is a particular solution and w 0(r,θ) is the solution of the homogeneous equation.
At this stage we introduce the normalized variables ρ = ρ/a and β =b/a. Given the geometry of the problem, the solutions will be different for the inner (ρ ≤ β) and the external (ρ > β) parts of the plate. As in related DM designs [7, 8], the relevant solutions are those for the inner part, so we will concentrate our analysis on them only. The particular solution in this region can be taken in the form
which corresponds to well known case of a circular plate simply supported along its boundary with a homogeneously distributed total load P over a ring of radius b, with zero deflection at the center of the plate, and upwards deflection is defined positive. This is equivalent to an edgesupported deformable mirror being pressed by an annulus of radius b [7, 8], showing a simple quadratic dependence on ρ.
The homogeneous equation must be solved simultaneously for both parts of the plate. The general solution for the external part is given by 
A similar solution is taken for the inner part of plate, but avoiding the terms that cause infinite deflection at the center, resulting in
where Ŕ0 = Á0+B́0ρ2 and Ŕm = Ámρm+Ćmρm+2. Therefore, for each term m in Eqs. (3) and (4) we have to determine four constants for the outer portion and two for the inner portion of the plate. These are obtained from the boundary conditions along the edge and from continuity along the circle of radius b. For this purpose, we also represent in the form of a series the two point forces at (ρ = b;θ = 0,π),
and their corresponding reactions at (r = a;θ = ±π/2),
Since the edge of the plate is only supported at two points, the boundary conditions along the external perimeter are written as 
Along the dividing circle we have the continuity conditions for the deflection,
The last boundary condition comes from the consideration of the shearing force along the circle ρ = β, continuous at all points except where the concentrated forces Fb(θ) are applied, expressed by the equation
Finally, using the superposition theorem, the deflection of the plate inside the region ρ < β is given by w i(r,θ) = w 1,i(ρ)+w 0,i(r,θ).
We will now perform an analysis of this solution in terms of Zernike polynomials. For this purpose, it is enough to consider the first term (m = 2) of the series, since each of the higher order terms as well as their total contribution are typically two orders of magnitude smaller than this term. We can therefore write the total deflection as
Let us a consider a flat wavefront normally incident on a mirror deformed according to the equation above. Upon reflection, the wavefront will acquire a deformation given by twice this function. In terms of the relevant Zernike polynomials , such a wavefront can be written as
where = ρ/β is now the radius normalized to the considered aperture and ai are the coefficients for the following polynomials:
The first polynomial corresponds to defocus, the second to astigmatism, and the third to a fifth-order aberration. By equalling the right hand sides of equations 12 and 13, and apart from the constant factor in Z 3( ) that does not contribute to the shape of the deflection, we obtain the solutions for the wavefront Zernike coefficients in terms of the deflection parameters,
From this, we can arrive at the important result that the ratio between the aberration coefficients of the reflected wavefront is independent of the applied force P or the mirror dimensions, depending only on the mirror material (through v) and the relative positioning of the actuators (through β). Figure 2 shows the weight of a 3 and a 11 relative to a 4 as a function of the parameter β. It is clear that by changing this parameter one can tune the ratio between defocus and astigmatism. In fact, this ratio varies in a smooth, almost linear fashion between β ≈ 0.2 (for which |a 3/a 4| ≈ 0.5) and the limit case β = 1, for which no defocus is introduced and the wavefront is purely astigmatic. In this configuration, both the particular solution (Eq. (1)) and the terms in Eq. (11) of the form m = 4,8,12… vanish, and the DM shape becomes independent of the material (i.e. of v), corresponding to a design with the point forces being applied at the edge, such as that described in Ref. . We can also observe that the role of the fifth-order aberration term a11 is negligible over the considered range, being two orders of magnitude lower than simple astigmatism. For β < 0.2 the gradient of the ratio |a 3/a 4| increases sharply and using this mirror becomes impractical. However, as we will see, as β approaches zero the analytical solution is not a valid description anymore, since the internal shearing stresses, which are not considered in this pure bending model, become relevant.
In Fig. 3 is shown a calculated wavefront profile and its corresponding interferogram for the case of a 75 mm diameter, 6 mm thick BK7 mirror with β = 0.57, a value that will also be used in the experimental characterization, for comparison purposes. The mirror parameters are the same as those used below. In order to evaluate the working range of the mirror, we have performed a stress analysis by using the expressions for the radial and tangential stresses,
Figure 4 shows the results for the same parameters. Analyzing the stress distribution as a function of β , we have found that the peak values occur for σθθ at the positions of the inner actuators, clearly identifiable as the red regions in the figure. Given the limiting tensile stress of the substrate (51.7 MPa for BK7), this imposes a maximum achievable deflection for each range of parameters. Figure 5 shows the dependence of the peak-to-valley deflection for three mirror thicknesses. We can see that for low values of β, where the defocus term becomes strong, leading to an almost cylindrical shape, the deflections are limited to a few microns. As β increases and the astigmatism dominates, deflections of tens of microns are possible. Since for real systems the required corrections are normally of a few wavelengths, in general the mirror will operate well below the limiting conditions. We can also conclude that using a thinner mirror allows larger deflections, since the stresses are reduced for an identical deformation.
3. Experimental characterization
In order to evaluate the performance of a deformable mirror with these characteristics, we designed an adequate mirror mount, with a single adjusting mechanism, as illustrated in Fig. 6. An off-the-shelf Ø75 mm, 6 mm thick, λ/10 dielectric mirror is enclosed in an external frame, and its deformation is controlled by moving a rear piston by means of a sub-micron precision differential adjuster (Thorlabs DM22). The front (coated) surface of the mirror is supported at two diametral points near its edge (a = 35 mm) by means of a pair of Ø2 mm, 9 mm long hard needle rollers, inserted on the inner side of a retaining ring and perpendicular to the mirror surface. A similar pair of needle rollers is mounted on the piston surface, pushing against the back surface of the mirror at two points along a perpendicular diameter. The piston surface was fitted with holes for the actuators at ±10, 15, 20, 25 and 30 mm, in order to allow variation of the parameter β. This configuration with discrete actuators has the advantage that no special requirements on the smoothness of the needle roller tips or on the evenness of their relative placement are required, since the mirror will automatically adjust itself through tip and tilt in order to provide mechanical equilibrium between the four actuators. For the frame and the piston we used 12 mm thick aluminum alloy, which provides a good compromise between lightweightness and mechanical characteristics. From the analytical model we estimated that a force of ~5 N per actuator would be required for producing a purely astigmatic deformation of ~1 μm P-V, which is compatible with the chosen micrometer adjuster. It is worthwhile noting that in this particular configuration the defocus introduced corresponds to that of a convex mirror, i.e. an incident flat wavefront will be divergent and astigmatic upon normal reflection. This is the most relevant case, since it can be used to compensate the positive curvature introduced by thermal lensing. A different design providing positive (concave) curvature can be achieved by considering a DM frame where the back actuators are mounted near the edge of the piston at radius a, and the retaining ring has a smaller aperture, so that the front actuators can be mounted at radius b. Since we are only considering the region delimited by the innermost actuator radius, this configuration introduces little or no loss in the usable clear aperture. Additionally, we have designed our DM frame so that it can be readily mounted on a conventional Ø75 mm mirror mount, providing continuous adjustment of the astigmatism angle by simply rotating the frame.
The DM setup was oriented with the back actuators along the vertical position, and placed at one arm of a Michelson interferometer. A CW beam from a Ti:sapphire oscillator operating at 1053 nm (Coherent Mira) was spatially filtered and enlarged by using a 20x objective, a 25 μm pinhole and a f = 60 cm collimating lens. A Ø10 cm dielectric beamsplitter was used for dividing and recombining the beam, and a Ø4 cm aperture placed after it was imaged on a CCD camera. Figure 7 shows a picture of the interferometer setup, illustrating the compact size of the DM assembly. The interferometer was first adjusted without introducing any deformation to the mirror to ensure that any residual aberration present was below the level for the quoted uniformity of all the optical surfaces (λ/10), and a reference interferogram was taken, which was then subtracted from all subsequent measurements. Interferograms were then taken for each position of the back actuators, and for a range of applied forces within each position. Deformations of several wavelengths were easily obtained without any visible damage to the mirror substrate or coating. The actual force acting on the mirror was not measured, since the chosen micrometer adjuster does not provide that information. We can however estimate its magnitude by comparing the measured deflections with those predicted by the analytical model. Based on this approach, we calculated that the maximum tested forces ranged between 90 N (for β = 30 mm) to 150 N (b = 10 mm). Finally, the obtained interferograms were processed using IDEA software .
4. Results and discussion
Figure 8 shows typical interferograms and the corresponding wavefront profiles for a range of values of β. The deviation from the fourfold symmetry typical of pure astigmatism clearly shows that there is a superposition of this aberration with defocus inside the circles delimited by the back actuators (shown in red for the first three cases). Figure 9 shows a 3D plot of the measured wavefront for the central result, and the residual wavefront after subtracting the corresponding Zernike polynomials given in Eq. (13) (for β = 0.57 we have a 3 = 0.0637,a 4 = -0.3677 and a 11 = 0.0099), with a RMS error of 0.076λ.
Since both a 3 and a 4 have a quadratic dependence on ρ, and according to the analytical model developed these aberrations are linearly superposed, we can use a simplified parameter for characterizing the DM performance, which is the ratio between the wavefront curvature along the horizontal (θ = π/2) and vertical (θ= 0) axes. This has the advantage that it is a dimensionless parameter, is also independent of the level of deformation applied or the mirror dimensions, and is experimentally easy to estimate by considering the number of fringes along each axis of the aberrated interferogram. Horizontal and vertical lineouts passing through the center of the several wavefront profiles were taken over a clear aperture of 0.9b , in order to avoid eventual localized border deformations. To this data a quadratic fit was made, and the quadratic term was taken as the corresponding curvature. The r 2 factor was generally above 0.99, showing that the induced deformation is indeed quadratic. This data was then compared with the results provided by the analytical model, where the corresponding curvatures were defined as ∂2 w i(ρ,θ)/∂ρ2 along θ = 0 and θ = π/2. Additionally, a simple 3D finite element analysis (FEA) code was used for simulating the geometry of the DM, and the same data was extracted. The results, shown in Figure 10, exhibit a remarkable agreement over most of the range of β , showing that the analytical model provides an adequate description. For low β a deviation between the analytical model and both the measured and FEA results becomes evident. As already mentioned, this is expected, since the analytical model is not adequate when the inner actuator radius is comparable to the mirror thickness and the internal shearing stresses are not negligible. However, that is only relevant for cases with large curvature ratios (corresponding to a defocus coefficient comparable to or larger than that of astigmatism), for which this specific kind of DM design is probably not the best choice either.
In conclusion, we have demonstrated that a four-point loaded DM is an excellent choice for simultaneous correction of astigmatism and defocus, with the balance between both aberrations being given by the relative position of the actuators. We have assembled and characterized a compact and inexpensive DM and showed that it performs according to the developed analytical model over the relevant range of parameters. This mirror design can be easily integrated into existing setups, and by reversing the actuator positions it is able to compensate for either convergent or divergent wavefronts. Although in our setup the choice of back actuator positions was discrete, it is easy to implement a derived mirror with two adjusting mechanisms for faster and precise correction of both aberrations.
This work and two authors (J. Wemans and L. Cardoso) are supported by Fundação para a Cieˆncia e a Tecnologia. The authors are thankful to Steven Hawkes and Trevor Winstone at the Rutherford Appleton Laboratory, UK, for useful discussions on the implementation of DM technology.
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